Apply function to pandas row-row cross product

You also can use pd.DataFrame constructor with apply:

pd.DataFrame(index=df2.squeeze(), columns=df1.squeeze()).apply(lambda x: x.name.astype(str)+':'+x.index)

Output:

            1        2        3        4                                        
one      1:one    2:one    3:one    4:one
two      1:two    2:two    3:two    4:two
three  1:three  2:three  3:three  4:three
four    1:four   2:four   3:four   4:four

Explanation:

First, with pd.DataFrame constructor, first build and empty dataframe with index and columns defined from df2 and df1 respectively. Using pd.DataFrame.squeeze, we convert those single column dataframes into a pd.Series.

Next, using pd.DataFrame.apply, we can apply a lambda function which adds the strings from the column name with a colon and the dataframe index for each column of the dataframe.

This yeilds a new dataframe with indexing and desired values.

Let us try np.add.outer

df = pd.DataFrame(np.add.outer(df1[0].astype(str).values,':'+df2[0].values).T)
Out[258]: 
         0        1        2        3
0    1:one    2:one    3:one    4:one
1    1:two    2:two    3:two    4:two
2  1:three  2:three  3:three  4:three
3   1:four   2:four   3:four   4:four